Weierstrass Substitution Worked Examples Cheat Sheet

The Weierstrass substitution rewrites trigonometric integrals using $t = \tan\frac{x}{2}$ . This changes many integrals involving $\sin x$ and $\cos x$ into rational functions of $t$ . A worked examples cheat sheet helps students recognize when the substitution is useful and apply the algebra without losing track of constants.

It is especially useful in college calculus when standard identities do not simplify an integral quickly.

The core formulas are $\sin x = \frac{2t}{1+t^2}$ , $\cos x = \frac{1-t^2}{1+t^2}$ , and $dx = \frac{2}{1+t^2}\,dt$ . After substitution, simplify the integrand into a rational expression and integrate using algebra, partial fractions, or basic antiderivatives. At the end, convert back using $t = \tan\frac{x}{2}$ unless the problem asks for an answer in terms of $t$ .

Key Facts

The Weierstrass substitution is $t = \tan\frac{x}{2}$ , which is also written as $t = \tan\left(\frac{x}{2}\right)$ .
The sine conversion formula is $\sin x = \frac{2t}{1+t^2}$ .
The cosine conversion formula is $\cos x = \frac{1-t^2}{1+t^2}$ .
The differential conversion formula is $dx = \frac{2}{1+t^2}\,dt$ .
The tangent conversion formula is $\tan x = \frac{2t}{1-t^2}$ when $1-t^2 \neq 0$ .
An integral of the form $\int R(\sin x,\cos x)\,dx$ becomes $\int R\left(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}\right)\frac{2}{1+t^2}\,dt$ .
For definite integrals, convert bounds using $t = \tan\frac{x}{2}$ , but watch for discontinuities when the interval crosses $x = \pi + 2\pi k$ .
After integrating in $t$ , substitute back with $t = \tan\frac{x}{2}$ to express the final answer in terms of $x$ .

Vocabulary

Weierstrass substitution: A trigonometric substitution using $t = \tan\frac{x}{2}$ to convert expressions involving $\sin x$ and $\cos x$ into rational functions.
Half-angle substitution: Another name for the substitution $t = \tan\frac{x}{2}$ because it uses the tangent of half the original angle.
Rational function: A function that can be written as a quotient of polynomials, such as $\frac{t^2+1}{t-3}$ .
Back-substitution: The step of replacing $t$ with $\tan\frac{x}{2}$ after finding an antiderivative in terms of $t$ .
Partial fractions: A method for rewriting a rational expression as simpler fractions that are easier to integrate.
Domain restriction: A limitation on where a substitution is valid, such as avoiding points where $t = \tan\frac{x}{2}$ is undefined.

Common Mistakes to Avoid

Forgetting to replace $dx$ is wrong because the substitution is incomplete without $dx = \frac{2}{1+t^2}\,dt$ .
Using $\sin x = \frac{t}{1+t^2}$ is wrong because the correct numerator is $2t$ , so the correct formula is $\sin x = \frac{2t}{1+t^2}$ .
Using $\cos x = \frac{t^2-1}{1+t^2}$ reverses the sign; the correct formula is $\cos x = \frac{1-t^2}{1+t^2}$ .
Changing definite bounds incorrectly gives the wrong area or accumulated value; each bound $x=a$ must become $t = \tan\frac{a}{2}$ .
Leaving the final indefinite integral in terms of $t$ can be incomplete when the original problem uses $x$ ; substitute back with $t = \tan\frac{x}{2}$ unless instructed otherwise.

Practice Questions

1 Use $t = \tan\frac{x}{2}$ to evaluate $\int \frac{dx}{1+\sin x}$ .
2 Use the Weierstrass substitution to evaluate $\int \frac{dx}{2+\cos x}$ .
3 Convert $\int \frac{\sin x}{1+\cos x}\,dx$ into an integral in $t$ using $t = \tan\frac{x}{2}$ , then simplify before integrating.
4 Explain why the Weierstrass substitution is often useful for integrals involving rational combinations of $\sin x$ and $\cos x$ .

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Weierstrass Substitution Worked Examples Cheat Sheet

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